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Minimal reduction type in classical cases

Bin Wang, Xueqing Wen, Yaoxiong Wen

TL;DR

The paper proves Yun's minimal reduction conjecture for all classical groups by showing that for any γ in the topologically nilpotent regular semisimple locus, RT_min(γ) is a single nilpotent orbit. It develops a unified framework that extracts the minimal reduction from the characteristic polynomial χ(γ) using Newton polygons and m-balanced partitions, and then extends the construction to the BD types through admissible partitions and compatible bilinear forms. It provides explicit, algorithmic procedures to compute RT_min(γ) from χ(γ), including both skeleton- and Newton-polygon-based approaches and careful pairing adjustments to handle very-even and parity constraints. The results extend Yun's A and C cases to B and D, connect with Lusztig's and affine Springer fiber perspectives, and have implications for related problems such as isoclinic Deligne–Simpson problems and the structure of affine Grassmannian strata.

Abstract

We prove Yun's minimal reduction conjecture for all classical groups. More precisely, for any topologically nilpotent regular semisimple element $γ$, we show that the associated minimal reduction set $\mathrm{RT}_{\mathrm{min}}(γ)$ consists of a single nilpotent orbit. This result confirms and extends Yun's earlier work in types A and C, and resolves the remaining cases in types B and D. Moreover, we provide an explicit and effective procedure for determining $\mathrm{RT}_{\mathrm{min}}(γ)$.

Minimal reduction type in classical cases

TL;DR

The paper proves Yun's minimal reduction conjecture for all classical groups by showing that for any γ in the topologically nilpotent regular semisimple locus, RT_min(γ) is a single nilpotent orbit. It develops a unified framework that extracts the minimal reduction from the characteristic polynomial χ(γ) using Newton polygons and m-balanced partitions, and then extends the construction to the BD types through admissible partitions and compatible bilinear forms. It provides explicit, algorithmic procedures to compute RT_min(γ) from χ(γ), including both skeleton- and Newton-polygon-based approaches and careful pairing adjustments to handle very-even and parity constraints. The results extend Yun's A and C cases to B and D, connect with Lusztig's and affine Springer fiber perspectives, and have implications for related problems such as isoclinic Deligne–Simpson problems and the structure of affine Grassmannian strata.

Abstract

We prove Yun's minimal reduction conjecture for all classical groups. More precisely, for any topologically nilpotent regular semisimple element , we show that the associated minimal reduction set consists of a single nilpotent orbit. This result confirms and extends Yun's earlier work in types A and C, and resolves the remaining cases in types B and D. Moreover, we provide an explicit and effective procedure for determining .
Paper Structure (33 sections, 50 theorems, 144 equations)

This paper contains 33 sections, 50 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Kazhdan--Lusztig map and the minimal reduction map
  3. Kazhdan--Lusztig map
  4. Minimal reduction map
  5. Main results
  6. Plan of the paper
  7. Notations
  8. Preliminary
  9. Minimal Reduction Map
  10. Nilpotent orbits
  11. Newton Polygons
  12. Pairings and Lie algebras in type B, C and D
  13. Companion Matrix and Pairings
  14. Construction of Minimal Reduction: Type A and C
  15. Construction via Skeleton
  16. ...and 18 more sections

Key Result

Theorem 1.3

Let $G$ be a classical group. For any $\gamma \in L^\heartsuit {\mathfrak g}$, the minimal reduction set $\operatorname{RT}_{\min}(\gamma)$ consists of a single nilpotent orbit. Moreover, this orbit can be determined explicitly from the characteristic polynomial $\chi(\gamma)\in L^+\mathfrak{c}$.

Theorems & Definitions (91)

  • Definition 1.1: Minimal reduction
  • Conjecture 1.2: Yun Yun21
  • Theorem 1.3
  • Definition 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 81 more